CHARLOTTE — After being pummeled for days at the Republican National Convention for his remark that business owners “didn’t build that,” President Obama heads to the Democratic National Convention in North Carolina this week facing mounting questions about how he will respond to charges that he is hostile to free enterprise.

On Sunday, senior Obama advisers suggested that they will not address the anti-business allegations directly but will instead try to turn the tables on their GOP rivals by accusing them of being dishonest about what Obama meant. David Plouffe, a senior White House adviser, said in an interview Sunday on ABC News that Republican Mitt Romney’s campaign is engaged in a broader pattern of dishonesty and is “built on a tripod of lies.” Plouffe cited accusations that Obama has gutted the work requirement for welfare and “raided” Medicare to pay for the nation’s new health-care law as other examples of untruths coming from the GOP.

The Obama team thinks that it has effectively dealt with the “build that” attacks and that the issue is overblown — the “drill, baby, drill” of 2012, a rallying cry for the right but ultimately one with limited appeal in the broader electorate.

Nevertheless, there are signs that they see a vulnerability. Obama has not repeated the words that sparked the controversy, and he has toned down the broader argument — that government help is essential to business success — in the six weeks since he ad-libbed the line near the end of a long campaign swing. His speeches have been shorter, with fewer references to wealthy Americans. He is more cautious about portraying the choice that he quite forcefully described that night between Romney’s worldview and his own.

Adviser David Axelrod, traveling with the president in Colorado on Sunday, said the public will come away from the convention “with a very clear sense” of Obama’s values, including his faith in private enterprise.

“It’s striking to me how enamored they are with that theme and how ineffectual it’s been,” Axelrod said. “You look at the polling and they’ve spent millions and millions of dollars on it and it may thrill their base. But it hasn’t expanded their base because people understand that the view they’re imputing to the president isn’t his view. I don’t feel like we have to respond to their dry holes.” Obama campaign advisers say internal polling shows that the GOP attacks have not shifted public opinion.

The “build that” accusations reached a fever pitch last week at the Republican convention.

Obama made the comment in July in Roanoke, saying: “If you were successful, somebody along the line gave you some help. There was a great teacher somewhere in your life. Somebody helped to create this unbelievable American system that we have that allowed you to thrive. Somebody invested in roads and bridges. If you’ve got a business — you didn’t build that. Somebody else made that happen.”

Republicans have typically quoted only the last part — “If you’ve got a business, you didn’t build that” — prompting most independent fact checkers to conclude that the line was taken out of context. That’s Obama’s argument, too; his advisers cite numerous speeches with similar language about the important role government plays paying for roads and bridges and other infrastructure to help businesses grow and prosper.

Republicans say that, even in context, it’s not clear whether Obama is referring to businesses or infrastructure when he states, “you didn’t build that.” They say the overall speech reinforces a narrative about Obama — that he places too much faith in government — that resonates with voters.

“We need a president who will say to a small businesswoman, ‘Congratulations!’ ” said Virginia Gov. Robert F. McDonnell (R), one of several speakers to cite the line in Tampa.

The tactic has certainly resonated with the Republican grass roots. Throughout the GOP convention, and even sporadically at Obama rallies across the country, activists have displayed “I Built That” T-shirts and signs. On Sunday, a huge banner on an airplane hangar greeted the president in Sioux City, Iowa, where the airport code is SUX: “Obama Welcome to SUX. And We Did Build This.”

During the original 42-minute speech in Roanoke on July 13, Obama used no teleprompter, instead relying on notes and at times injecting lengthy and impromptu riffs about the role government has played in building this nation. He talked of an elderly veteran who relied on the G.I. Bill to go to college and a single mother who got an education with grants. He criticized Republicans for wanting to cut taxes for the wealthiest Americans at the expense of government programs that benefit the middle class. But in this instance, he lingered on the point.

He blamed the GOP for what he described as undermining the American contract that allows all people to succeed if they’re willing to work hard. He talked of “rich people,” “millionaires” and “wealthy investors.” At times he seemed to lecture his audience about the stark choice he sees in this election, waiting for the crowd to quiet so he could continue.

Obama advisers say the president did not seek to make news that night in Roanoke, and that his message has grown sharper since then because that’s what happens over the course of a campaign. They say the speech came at the end of a long day — he did appear tired, and his voice was hoarse — and say if he’d intended to try out a new message, it would have happened at his first stop of the tour.

That evening, Obama spoke nearly twice as long as he has in more recent campaign rallies — a tendency, his advisers said, when he is fatigued. It was a hot, sticky night; more than 20 people in the crowd required medical attention, and at one point, even Obama noticed from the stage, advising supporters to “make sure you’re drinking water.”

Republicans seem to have noticed that Obama was tired, too. “They’re running this guy ragged,” GOP strategist Karl Rove said in an interview, adding that the president’s “normal filters shut down.” The implication was that the GOP will be watching for more such instances for the duration of the campaign.

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